MORSE THEORY AND TILTING SHEAVES DAVID BEN-ZVI AND DAVID NADLER 1. Introduction This paper is a result of our trying to understand what a tilting perverse sheaf is. (See [1] for definitions and background material.) Our basic example is that of the flag variety B of a complex reductive group G and perverse sheaves on the Schubert stratification S. In this setting, there are many characterizations of tilting sheaves involving both their geometry and categorical structure. In this paper, we propose a new geometric construction of such tilting sheaves. Namely, we show that they naturally arise via the Morse theory of sheaves on the opposite Schubert stratification Sopp. In the context of a flag variety B, our construction takes the following form. Given a sheaf F on B, a point p ∈ B, and a germ F of a function at p, we have the ∗ corresponding local Morse group (or vanishing cycles) Mp,F (F) which measures how F changes at p as we move in the family defined by F . If we restrict to sheaves F constructible with respect to Sopp, then it is possible to find a sheaf ∗ Mp,F on a neighborhood of p which represents the functor Mp,F (though Mp,F is not constructible with respect to Sopp.) To be precise, for any such F, we have a natural isomorphism ∗ ∗ Mp,F (F) ' RHomD(B)(Mp,F , F). Since this may be thought of as an integral identity, we call the sheaf Mp,F a Morse kernel. The local Morse groups play a central role in the theory of perverse sheaves. For an arbitrary sheaf on Sopp, its local Morse groups are graded by cohomological degree. The perverse sheaves on Sopp are characterized by the fact that their local Morse groups are concentrated in degree 0. In other words, the local Morse groups are t-exact functors (with respect to the perverse t-structure), and any sheaf on which they are concentrated in degree 0 is in turn perverse. Now choose a one-parameter subgroup C× ⊂ G so that for the resulting C×- action on B, the strata of S are its ascending manifolds, and the strata of Sopp are its descending manifolds. In other words, the action has finitely many fixed points, and for z ∈ C×, the action of z contracts the strata of S to the fixed points as z → ∞, and contracts the strata of Sopp to the fixed points as z → 0. Then Date: June 23, 2005. 1 2 DAVID BEN-ZVI AND DAVID NADLER × we may take a Morse kernel Mp,F and flow it down along the C -orbits to obtain a limit sheaf (or nearby cycles) z→0 Mp,F −→ Ψ(Mp,F ). The limit is a sheaf on B with, a priori, no particular shape. Our main result is the following. (See [6] for another application of such limits.) Theorem 1.1. For a Morse kernel Mp,F , the shifted limit Ψ(Mp,F )[dimC B] is a tilting perverse sheaf on the Schubert stratification S. For a sheaf F on the opposite Schubert stratification Sopp, we have a natural isomorphism ∗ ∗ RHomD(B)(Mp,F , F) ' RHomD(B)(Ψ(Mp,F ), F). It is easy to see that all tilting sheaves on S arise as summands of such limits. Thus we think of the result as giving a geometric construction of tilting sheaves. We mention a possible source of confusion. It is well known that such limits (or nearby cycles) take perverse sheaves to perverse sheaves. But the Morse kernels are by no means perverse (they are not even C-constructible.) Nevertheless, the above theorem asserts that by taking their limits along the C×-orbits we obtain perverse sheaves. To connect with traditional notions of representability, consider the opposite unipotent subgroups N, N opp ⊂ G whose orbits in B are the strata of S, Sopp respectively. Since N, N opp are contractible, the N, N opp-equivariant derived cat- egories are the same as the S, Sopp-constructible derived categories respectively. Thus we have the equivariantization functors + − Γ : D(B) → DS(B) γ : D(B) → DSopp (B) which are right and left adjoint respectively to the forgetful functor. If we re- strict Γ+, γ− to Sopp, S-constructible sheaves respectively, then they become in- verse equivalences of categories. Using our main theorem, we see that for a Morse kernel Mp,F , the equivarianti- − − zation γ commutes with the limit Ψ. It follows that γ (Mp,F ) is a projective perverse sheaf on Sopp. Thus we have a triangle of objects which represent the local Morse group Mp,F Ψ .& γ− γ− opp Tp,F [− dimC B] −→ Pp,F At the top, we have the Morse kernel Mp,F which by construction represents Mp,F . To the left, we have the shifted tilting sheaf Tp,F [− dimC B] on S to which Mp,F flows. By the above theorem, Tp,F [− dimC B] also represents Mp,F . Finally, opp opp to the right, we have the projective perverse sheaf Pp,F on S which represents Mp,F both in the abelian and derived categories. MORSE THEORY AND TILTING SHEAVES 3 As a consequence, we see that the equivariantization functors exchange the shifted minimal tilting and projective perverse sheaves on the opposite stratifica- tions − opp + opp γ (Tw[− dimC B]) ' Pw Γ (Pw ) ' Tw[− dimC B] (Here the subscript w labels the Schubert stratum corresponding to w in the Weyl group of G.) This statement should be compared to the fact (see [1]) that for perverse sheaves on S alone, the “longest interwiner” exhanges the minimal tilting and projective perverse sheaves. The above statement is closely related to this, but continues to make sense in other contexts such as the affine setting. (This will be developed in [5].) 1.2. Organization of paper. In what follows, we work in a more general context than discussed in the introduction above. Namely, we do not restrict ourselves to flag varieties, but rather consider any space with the necessary properties for our main construction to be easily understood. This is explained in Section 2 along with other background material. In Section 3, we briefly recall the definition of perverse sheaves and tilting sheaves and their basic properties. In Section 4, we discuss Morse groups and Morse kernels on stratified spaces. Section 5 contains our main result and its proof. Finally, in Section 6, we return to the case of flag varieties. 1.3. Acknowledgments. We would like to thank Mark Goresky, Tom Nevins, and Kari Vilonen for useful comments. 2. Stratified spaces This section contains basic material concerning stratifications of manifolds: regularity conditions, Whitney stratifications, and normally nonsingular inclu- sions. We also recall the notion of a Bekka stratification [3] and describe the opposite stratifications to which our main theorem applies. 2.1. Stratifications. Definition 2.2. A stratification of a topological space X consists of a locally finite collection S = {Sα} of locally closed subspaces Sα ⊂ X called strata satisfying S (1) (covering) X = α Sα, (2) (pairwise disjoint) Sα ∩ Sβ = ∅, for α 6= β, c c c c c c (3) (axiom of frontier) Sα ∩Sβ 6= ∅ if and only if Sβ ⊂ Sα where Sα,Sβ denote connected components of Sα,Sβ respectively. We would like the group of stratification-preserving homeomorphisms of a space to act transitively on each of the connected components of its strata. In order for this to be true, we must impose some regularity on the stratification. Further- more, we would like whatever regularity we impose to behave well with respect to 4 DAVID BEN-ZVI AND DAVID NADLER common constructions such as forming products. There are many approaches to this problem which balance strength and applicability. In this paper, we follow the development in which differential topology plays the lead role. In particular, we assume that our topological spaces are manifolds and the strata are submani- folds. In order to keep technical issues at a minimum, we work with real analytic manifolds. Assumption 2.3. In this paper, unless stated otherwise, all manifolds and maps between them are assumed to be real analytic. By a stratification of a real an- alytic manifold, we mean a stratification in which the strata are real analytic submanifolds. Definition 2.4. Let S = {Sα}, T = {Tβ} be stratifications of manifolds M, N respectively. A map f : M → N is said to be stratified with respect to S and T provided that −1 (1) for each stratum Tβ, the inverse-image f (Tβ) is a union of connected c components Sα of strata Sα, c (2) if f maps a connected component Sα of a stratum Sα to the stratum Tβ, c c then the restriction f|Sα : Sα → Tβ is a submersion. 2.5. Whitney stratifications. Although we will work with an alternative ap- proach (described in the next section), we recall here the notion of a Whitney stratification for the sake of comparison. Let M be a manifold, let X, Y ⊂ M be disjoint submanifolds, and let x ∈ X∩Y . Choose local coordinates around x so that some neighborhood in M is identified with an open subspace of affine space. The following condition is easily seen to be independent of the choice of local coordinates. Definition 2.6 (B-regularity). The triple (Y, X, x) is said to satisfy Whitney’s condition B if given any sequences of points xi ∈ X and yi ∈ Y each converging to x, such that the secant lines `i = xiyi converge to some line ` and the tangent planes Tyi Y converge to some plane τ, we have ` ⊂ τ.